Here, we present the behaviour of DW dynamics in two different lengths i.e., \(L_{x}=1024\)  nm and \(L_{x}=2048\)  nm of biaxial nanowire in Fig. REF (a) and (b) respectively, as a function of \(\alpha \)  (ranging from 0.001 to 0.006) for fixed TG \(\nabla _{x}T = 0.5\)  \(\text{K}/\text{nm}\) . Fig. 4(a) shows the DW linear velocity as a function of \(\alpha \)  corresponding to the nanowire lengths, \(L_{x}=1024\)  nm (black) and \(L_{x}=2048\)  nm (red). In Fig. REF (b), the black line and red line show the change of angular velocity as a function of \(\alpha \)  corresponding to the lengths \(L_{x}=1024\)  nm and \(L_{x}=2048\)  nm respectively. DW angular velocities also decrease with \(\alpha \) . Since the magnons propagation length decreases with \(\alpha \)  [1]}, it reduces the number of magnons to pass through the DW, which leads to the reduction of DW velocity proportionately. It is noted that for smaller \(L_{x}=1024\)  nm (Fig. REF (a)), the DW speed is lower for \(\alpha = 0.001\) , and it gradually increases and then monotonically decreases. Interestingly, such behaviour disappears with the increase of length, and for the length \(L_{x}=2048\)  nm and the DW velocities show decreasing trend monotonically. The reason for observing such behaviour has been discussed above for shorter \(L_{x}=1024\)  nm. Here we mention again shortly that for lower damping, the magnon propagation length is larger. If the length of the sample size is not large enough, then there is a probability of forming the standing waves of some modes between the fixed boundaries. Consequently, the energy transfer through the DW is decreased, i.e., DW velocity is decreased because of the reduction of the magnonic current.
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